COMPRESSED SENSING 3 D Digitization Course Carlos Becker

COMPRESSED SENSING 3 D Digitization Course Carlos Becker, Guillaume Lemaître & Peter Rennert

OUTLINE • Introduction and motivation • Compressed sensing and reconstruction workflow • Applications: MRI and single-pixel camera 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 2

WHAT IS COMPRESSED SENSING? SIGNAL SPARSITY 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 3

WHAT IS COMPRESSED SENSING? WHY DO WE CARE ABOUT SPARSITY? Original 1 Megapixel image Non-sparse values 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 4

WHAT IS COMPRESSED SENSING? WHY DO WE CARE ABOUT SPARSITY? But, in the wavelet domain we get these coefficients: 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 5

WHAT IS COMPRESSED SENSING? WHY DO WE CARE ABOUT SPARSITY? And the histogram of those coefficients is: The image is a nearly sparse in the wavelet domain… 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 6

What happens if we only keep the 50000 highest coefficients in the wavelet domain, set the rest to zero and reconstruct the image ? Original image Reconstructed image (only 50 k highest wavelet coefficients) 95% of the original image data was discarded 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 7

WHAT IS COMPRESSED SENSING? • Classic approach to compression: – Measure everything (ie: all pixels) – Apply some compression algorithm (ie: JPEG 2000) Full-resolution image (N pixels/measurements) Lossy compression – But, why would we sample 1 million pixels if we are going to throw away 90% of image data when compressing the image in JPEG? • Compressed sensing approach: if signal is sparse in some domain – Sample M << N random measurements – Reconstruct original signal by L 1 minimization Random sampling (M << N measurements) 8/11/2010 Image reconstruction Candès et al. showed that it is possible to subsample a signal if it is sparse in some domain, being able to obtain a perfect reconstruction if certain conditions are met. Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 11

COMPRESSED SENSING MOTIVATION FROM MRI • • • 2004, Candes came to results that people of his time could not believe For a simple phantom (a) its possible to sample at only 22 radial lines (b) (equal to a sampling rate of π / 22, about 50 times below the Nyquist rate of 2 π) to retrieve a perfect reconstruction (d) What does the trick? Simply setting the unknown Fourier coefficients to 0 leads to a very bad result (c) Candès, E. J. ; Romberg, J. ; Tao, T. : “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information” (2004) 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 12

COMPRESSED SENSING RECONSTRUCTION WORKFLOW • • • Sparse signal gets randomly sampled in another non-sparse domain (k-space) Reconstruction leads to noisy nonsparse signal with significant peaks where original signal was high After thresholding of significant peaks the strongest components of the original signal are detected Using the noisy reconstruction of the newly sampled strongest components in k-space, the impact of this strongest components on the first reconstruction are determined and subtracted, leaving peaks of weaker components With this iterative strategy weaker and weaker components can be retrieved Michael Lustig, David Donoho, John M. Pauly: “Sparse MRI: The application of compressed sensing for rapid MR imaging” (2007) DL Donoho, I Drori, Y Tsaig : Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit” (2006) 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 13

COMPRESSED SENSING RAPID MRI – NON-SPARSE SIGNAL (Signal here means the underlying image that is sensed in the Fourier space) • • • Non- sparse signal sampled in sparse domain That means: reconstruction of samples will produce no significant peaks (since there are no outstanding peaks in the signal domain) Solution: use other sparse domain of signal for “reconstruction” and filtering of significant peaks Michael Lustig, David Donoho, John M. Pauly: “Sparse MRI: The application of compressed sensing for rapid MR imaging” 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 16

SINGLE PIXEL CAMERA GENERAL PRINCIPLE Object Photodiode Σ Several Measurements Memory Reconstruction DMD M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. G. Baraniuk, "Single-Pixel Imaging via Compressive Sampling, " IEEE Signal Processing Magazine, Vol. 25, No. 2, pp. 83 -91, March 2008 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 17

SINGLE PIXEL CAMERA RESULTS 2 % measurements 5 % measurements 10 % measurements 20 % measurements Original: 16384 pixels M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. G. Baraniuk, "Single-Pixel Imaging via Compressive Sampling, " IEEE Signal Processing Magazine, Vol. 25, No. 2, pp. 83 -91, March 2008 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 22

COMPRESSED SENSING CONCLUSION • Compressed sensing lets us sub-sample a signal w. r. t. Nyquist rate and reconstruct it perfectly, if this signal is known to be sparse in some domain and some conditions are met • Compressed sensing is promising for a wide range of future technologies, specially for high-frequency signals – Speeds up acquisition process: specially interesting for MRI – Cheaper hardware (ie: IR cameras with only a few sensors) • Sparsity can also be exploited for classification and image processing tasks[Huang, K. and Aviyente, S. , Sparse representation for signal classification] 8/11/2010 Compressed Sensing - Carlos Becker, Guillaume Lemaître, Peter Rennert 23